Let \({F}_B\)  be the set of the finite linear combinations of Laguerre functions endowed with the algebraic structure of \({L}^1_{B}\) .
Clearly \({F}_B\subset {S}_{B}\)  and Lemma REF  ensures that \({F}_B\)  is closed under product. In fact, one has that \({F}_B\)  is a \(\ast \) -subalgebra of \({L}^1_B\)  which is dense in view of the density of \({F}_B\)  in \(L^1\big (\mathbb {R}^2\big )\)  [1]}. In summary:
